# Math Help - Application of strong operator convergence

1. ## Application of strong operator convergence

Hello!

Any help with the following is really appreciated (hints, as well!):

For $f \in L^{1} ( \mathbb{R} )$ and $s \in \mathbb{R}$, let

$\hat{f} (s) = \int_{\mathbb{R}} f (t) e^{-ist} dt
$

Prove that $\hat{f} (s) \rightarrow 0
$
as $|s| \rightarrow \infty
$

We're allowed to use the 'strong operator convergence' theorem for Banach spaces, as well as other standard results in Banach spaces.

2. That is the Riemann-Lebesgue Lemma. You can find the proof on page 103 of "Papa Rudin" (Real and Complex Analysis). You can also go here for an outline of the proof.